MinT - Best Known (4, ∞, s)-Nets in Base 25

Best Known (4, ∞, s)-Nets in Base 25

(4, ∞, 66)-Net over F₂₅ — Constructive and digital

Digital (4, m, 66)-net over F₂₅ for arbitrarily large m, using

net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
  - a function field by GarcÃa and Quoos [i]

(4, ∞, 127)-Net over F₂₅ — Upper bound on s (digital)

There is no digital (4, m, 128)-net over F₂₅ for arbitrarily large m, because

m-reduction [i] would yield digital (4, 104, 128)-net over F₂₅, but
- extracting embedded orthogonal array [i] would yield linear OA(25¹⁰⁴, 128, F₂₅, 100) (dual of [128, 24, 101]-code), but
  - residual code [i] would yield linear OA(25⁴, 27, F₂₅, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,25)), but
    - bound for MDS codes [i]

(4, ∞, 130)-Net in Base 25 — Upper bound on s

There is no (4, m, 131)-net in base 25 for arbitrarily large m, because

m-reduction [i] would yield (4, 129, 131)-net in base 25, but
- extracting embedded OOA [i] would yield OA(25¹²⁹, 131, S₂₅, 125), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 53 976053 469340 278908 664699 142502 497319 475002 277726 758656 398146 688553 698769 765169 112321 921896 701801 416003 420587 163435 397481 219368 417699 666835 331273 606612 967341 789044 439792 633056 640625 / 21 > 25¹²⁹ [i]